Dilemma at RF Energy Harvesting Relay:
Downlink Energy Relaying or Uplink Information
Transfer?
Deepak Mishra, Swades De and Dilip Krishnaswamy
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Mishra, D., De, S., Krishnaswamy, D., (2017), Dilemma at RF Energy Harvesting Relay: Downlink Energy Relaying or Uplink Information Transfer?, IEEE Transactions on Wireless Communications, 16(8), 4939-4955. https://doi.org/10.1109/TWC.2017.2704084
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IEEE.
Dilemma at RF Energy Harvesting Relay: Downlink
Energy Relaying or Uplink Information Transfer?
Deepak Mishra, Swades De, and Dilip Krishnaswamy
Abstract—Performance of RF powered communication net-work is bottlenecked by short downlink energy transfer range and doubly-near-far problem faced in uplink information trans-fer to Hybrid Access Point (HAP). These problems can be resolved by cooperation of an RF energy harvesting node R present between HAP and RF energy harvesting information source S. However, there lies a dilemma at R on whether to transfer its harvested energy to S or to act as an information relay for transferring its data to HAP in a two-hop fashion. This paper resolves this dilemma at R by providing insights on its optimal positions suited for either energy relaying (ER) or information relaying (IR). It also investigates the possibilities of integrated ER and IR along with the regions where neither ER nor IR will be useful. In this regard, while considering Rician fading channels and practical nonlinear RF energy harvesting model, the expression for mean harvested dc power at S via energy transfer from HAP and ER from R is first derived. The closed-form outage probability expression is also derived for decode-and-forward relaying with maximal-ratio-combining at HAP over Rician channels. Using these expressions insights on optimal relaying mode is obtained along with global-optimal utilization of harvested energy at R for ER and IR to maximize the delay-limited RF-powered throughput. Numerical results validate the analysis and provide insights on the optimal relaying mode.
Index Terms—Integrated information and energy relaying, practical RF energy harvesting model, Rician fading, outage analysis, throughput maximization, generalized convexity
I. INTRODUCTION
Radio frequency (RF) energy transfer (ET) has drawn wide recent attention due to its capability of providing controlled energy replenishment of low-power wireless devices. Unlike the inductive and magnetic resonant coupling based non-radiative ET approaches, non-radiative RF-ET bestows benefits [2] like relaxed node-alignment requirements, beamforming capa-bility, and possibility of transmitting both energy and informa-tion over the same signal. This has led to the emergence of two attractive solutions for powering next generation networks: (a) Wireless Powered Communication Network (WPCN) [3] and (b) Simultaneous Wireless Information and Power Transfer (SWIPT) [4]. In WPCN, uplink information transfer (IT) is powered by downlink ET from Hybrid Access Point (HAP), whereas in SWIPT both ET and IT occur in same direction.
Despite the merits of RF-ET, there are some bottlenecks in its widespread usage. Major challenges [5] include wireless D. Mishra and S. De are with the Department of Electrical Engi-neering, Indian Institute of Technology Delhi, New Delhi, India (e-mail: {deepak.mishra, swadesd}@ee.iitd.ac.in). D. Krishnaswamy is with IBM India Research Laboratory, Bangalore, India (e-mail: dilikris@in.ibm.com).
This work was supported by the Department of Science and Technology (DST) under Grant SB/S3/EECE/0248/2014 and by the 2016-2017 IBM PhD Fellowship Awards program. A preliminary version [1] of this work has been accepted for presentation at the IEEE ICC 2017.
propagation and energy dissipation losses, low energy sen-sitivity, low rectification efficiency at low input power, and doubly-near-far problem [6] in WPCN. Therefore, investiga-tion of new paradigms is needed for efficient WPCN operainvestiga-tion.
A. Related Art
Various aspects of cooperative relaying have got recent research attention [5]–[20] to overcome doubly-near-far pro-blem [21] in WPCN and large difference between energy
and information sensitivities (−10 dBm versus −60 dBm)
in SWIPT. The authors in [5]–[10], [16]–[20] considered energy harvesting relay where an energy constrained node uses the harvested energy for cooperation. Optimal RF harvesting energy relay placement was investigated in [5] for maximizing the received power in two-hop RF-ET, both with and without distributed beamforming. In [6], a nearby node to HAP was considered to act as energy harvesting information relay for the farther node. A harvest-then-cooperate protocol was proposed in [7], where the relay node close to HAP harvests energy during downlink ET from HAP and then uses this energy for uplink information relaying (IR). The authors in [19] investigated instantaneous and delay-constrained throughput maximization for RF-powered full-duplex MIMO relay system by designing receive and transmit beamformers while optimi-zing the time-splitting parameter. A three-node RF-powered relay system was studied in [20] to maximize the ergodic throughput by optimizing the mode switching rule and transmit power jointly under the data and energy causality constraints. Different from [6], [7] which considered fixed relaying, the approach in [8] dynamically decides whether the nearby node should act like an information relay for far node or not. Further in [9], the roles as source, destination, or relay, for the nodes were dynamically decided. Optimal allocation of harvested energy at relay, due to SWIPT from multiple sources for forwarding data to their respective receivers, was considered in [10]. In [16] a greedy protocol was proposed for switching between energy harvesting and data relaying to minimize outage probability in amplify-and-forward (AF) energy harvesting relay network without direct link. In [17] this work was extended to distributed multi-relay selection with decode-and-forward (DF) two-hop IT. More recently, a relay selection scheme, incorporating channel conditions and battery status, was proposed in [18] to choose one among mul-tiple AF energy harvesting relays for IT. Furthermore, energy cooperation and sharing strategies have been proposed in [11], [12] to overcome dynamics of ambient energy harvesting and enable perpetual operation. In another set of works [13]–[15],
Phase 2:
Inf ormation
Transfer
Phase 2: Information Transfer
Phase1(a): Ener
gy Transfer Phase 3: Information Relaying Phase 1(b): Energy Relaying PtA β(1 − α)E h R βαE hR PtS PtS R
A Hybrid AccesPoint (HAP) S
Energy Harvesting Relay Energy Harvesting Information Source Information Link Energy Link
Fig. 1:Network topology for the RF-powered integrated 2-hop full-duplex energy relaying and half-full-duplex decode-and-forward IR.
relay-powered communications was considered, where energy-sufficient relay transfers energy to RF harvesting nodes. B. Motivation and Contributions
In recent studies, harvested energy at relay is either used for energy relaying (ER) [5] or IR [6]–[10], [16]–[20]. Though [13]–[15] studied tradeoff in ET and IR efficiency assuming energy-rich relay, these works along with [11], [12] did not investigate RF harvesting relay assisted ER possibilities. This work fills this existing research gap. It studies optimal utiliza-tion of harvested energy at RF-energy harvesting (EH) relay for ER and/or IR to enhance the performance of two-hop RF-powered delay-limited network with direct link availability. Key contributions of this work are six-fold: (1) As shown in Fig. 1, a novel system model is presented to investigate the per-formance of RF-powered integrated information and Energy
Relaying (i2ER) in WPCN (Section II). (2) Mean harvested
energy due to full-duplex ER over Rician fading channels with distributed beamforming is derived while considering practical RF harvesting model (Sections III). (3) Closed-form expressions for outage probability and normalized throughput are obtained for half-duplex DF-IR over Rician channels with Maximal Ratio Combining (MRC) (Section IV). (4) Analytical insights on optimal mode selection policy at RF harvesting relay are provided (Section V). (5) Global-optimal utilization of harvested energy at EH relay is obtained by efficiently solving the nonconvex delay-limited throughput maximization problem (Section VI). (6) Numerical results validate the ana-lysis and give insights on optimal harvested energy utilization in ER and IR for varying relay positions (Section VII). C. Novelty and Scope
To the best of our knowledge, this is the first work that
considers i2ER in WPCN and resolves the “dilemma” at RF
harvesting relay on whether to perform ER, IR, or i2ER by
jointly optimizing cooperation in ET and IT. We also present novel analyses on full-duplex ER with distributed beamforming and half-duplex DF-IR with MRC over Rician channels.
Results presented in this paper demonstrate the importance of i2ER, because in static node deployment scenarios where the relay position is fixed or the set of available routers are known, mode selection (ER or IR) becomes critical. This work
providing insights on optimized mode of cooperation with RF harvesting relay (for energy, information, or both) can be extended to multi-node scenarios, allowing IR on one path and ER on other for greater end-to-end efficiency depending on the position of the relays. Energy beamforming [3] can also be ap-plied over the proposed optimized cooperation for further en-hancement of achievable gains. Though the widespread utility of the proposed system architecture is constrained by the low RF-ET range [2], [5], there are some practical applications that can benefit from this proposal. These include low power EH nodes in small cell networks, miniature RF-powered sensing devices for indoor applications, and EH nodes in Internet-of-Things. Furthermore, with the advancement in RF-EH circuits technology [22], [23], this limited end-to-end RF-ET range will be significantly increased due to improvement in both RF-to-dc rectification efficiency and receive energy sensitivity. Another attractive solution to improve the performance of the proposed RF-powered i2ER is by implementing the
full-duplex IR. However this improvement comes at the expense of implementing loopback interference suppression with the help of sophisticated electronic schemes or spatial domain precoding techniques that require perfect channel estimation. So there is a need for low-cost energy-efficient full-duplex IR techniques for RF-powered relaying systems.
II. SYSTEMMODEL
Here we present the i2ER system model that includes
transmission protocol, network topology, and wireless channel, along with the energy consumption and RF harvesting models. A. Integrated Information and Energy Relaying in WPCN
We consider RF-EH relay R assisted full-duplex two-hop
downlink ET to RF-EH information source S and DF
half-duplex uplink IT to HAP A (cf. Fig. 1). We assume that
A and S are composed of single omnidirectional antennas, whereas R has two directional antennas; one pointing in the direction ofA – essentially for efficient EH at R and effective IR from R-to-A, and other directed towards S for efficient
downlink ER and improving the quality of S-to-R uplink
IT. Although half-duplex IR can be conducted using a single
omnidirectional antenna at R, two directional antennas are
considered to minimize the dissipation losses in downlink ER and uplink IR. Also, this helps in implementing two-hop full-duplex ER because RF energy signals do not contain any information to be lost during full-duplex operation. The role of full-duplex relaying in ER phase is to ensure that R
can simultaneously harvest energy from A as well transfer
energy toS in the Nsth slot. Moreover, as the dimensions of
the directional and omnidirectional antennas are similar [24] and the storage capacity of commercially available efficient compact supercapacitors [25] is large enough to store the harvested energy over multiple slots Ns, the form factor and
storage capability are not a concern in the proposed i2ER
architecture. The entire i2ER process can be divided into three
phases, as highlighted in Fig. 1 and Table I:
(1) RF-ET over Nsslots:Apart form single-hop ET fromA
TABLE I:Description of operations in RF-powered communication with energy and information relaying possibility.
Operations in RF po wered tw o-hop ET and DF-IT
RF Energy Transfer Information Transfer A to S and R Phase 1(a) R to S Phase 1(b) S to A and R Phase 2 R to A Phase 3
Duration Nsslots 1 slot 1 slot 1 slot
Slots # 1, 2, . . . , Ns Ns Ns+ 1 Ns+ 2 Relaying mode No relaying X X X X Energy X X X X Information X X X X i2ER X X X X
also involves two-hop full-duplex ET fromR to S using βα fraction of energy harvested EhRatR over Nsslots. (2) IT from S to A and R during (Ns+ 1)th slot using its
energy harvested over Nsslots.
(3) IT from R to A in (Ns+ 2)th slot using β (1− α)
fraction of harvested energy EhR atR.
Following the summary given in Table I, we note that the four possible relaying modes are: no relaying (NR), ER, IR, and i2ER. In NR (i.e., neither ER nor IR) mode, indicator variable β is set to zero. β = 1 for other modes represents thatR uses α fraction of its harvested energy EhR for ER and remaining fraction for IR. Thus α = 1 and α = 0 respectively represent ER and IR scenarios, whereas 0 < α < 1 represents the i2ER mode. Here it is worth noting that, since the energy harvested over a single slot is very low (as shown later in Fig. 3), we have considered Ns> 1 slots for RF-ET because
it helps in incorporating the hardware limitations of RF-EH communication [2], [5] while giving insights on the practically realizable rate-constrained sustainable throughput performance (cf. Section VII). Also, in order to have sufficient harvested energy atR for efficient RF-ET to S, ER is considered only in the Nsth slot. Further, due to the usage of two directional
antennas at R, the leaked energy from R in the unintended
direction, that could be recycled by its receiver, is neglected. B. Network Topology
To avoid blocking of direct ET and IT paths between A
andS, we consider parallel topology [5] for relay placement.
Considering static node deployment, A and S are
respecti-vely located on Euclidean plane with coordinates (0, 0) and (dAS, 0). Here dAS is the distance between A and S. R is
positioned at xR, yR0, where yR0 is the minimum
non-blocking distance [5] from Line-of-Sight (LoS) path between A and S. However, it is worth mentioning that the proposed
i2ER model and the analysis hold for any arbitrary relay
placement topology ensuring the availability of unaffected direct link betweenA and S, e.g., elliptical topology [13]. C. Channel Model
All the links are considered independent and experience flat quasi-static Rician block fading [13], where the average channel power gains hij
2 = Eh hij 2i = GiGj (dij)n 4πλ 2 ,
∀i, j = {A, R, S}. Here Gi and Gj represent antenna gain
for transmitting node i and receiving node j; λ is the wa-velength of transmitted RF signal; n is path loss exponent;
dij corresponds to distance between nodes i and j. Rician channel model helps in incorporating the effect of strong LoS component in RF-ET over short range IT links. Rician fading also includes Rayleigh fading [6]–[10] as its special case. To
reduce signalling overhead at energy-constrained R and S,
we assume that knowledge about statistics of channel state information (CSI), instead of instantaneous CSI, for all links is available at A via pilot signals received from R and S.
D. Energy Consumption and RF Harvesting Model
We assume thatS uses its entire harvested energy for IT. So with unit slot duration T = 1 s, energy or power consumption at S during IT is: PtS + P
tx
con, where Pcontx is static power
consumption independent of transmit power PtS. Generally Pcontx ≈ 0 in cooperative WPCN and SWIPT [6]–[10]. We note that RF energy reception does not consume any power [2], [5]. The consumption at R for IR is accounted as [26, Table I]: Erx
con+ R0Ebitrx, where E rx
con is static consumption and R0Ebitrx
is consumption in reception and decoding of R0 bits.
The harvested dc power Ph is a nonlinear function of
received RF powerPr[5]. To this end, we present a piecewise
linear approximation for establishing a relationship between Ph andPr using a piecewise linear functionL (·).
Mathema-tically,Ph=L (Pr) can be defined as:
Ph, 0, Pr<Pth1, MiPr+Ci, Pr∈Pthi,Pthi+1 , ∀ i=1, .., N, Not applicable, Pr>PthN +1, (1)
wherePth ={Pthi| 1≤i≤N +1} mW are thresholds on Pr
that define the boundaries for N linear pieces with slopeM =
{Mi | 1 ≤ i ≤ N} and intercept C = {Ci| 1 ≤ i ≤ N} mW.
We have used the above approximation L (·) because this
simple linear relationship betweenPrandPhhelps in gaining
insights on global-optimal harvested energy utilization and proving conditional-unimodality of throughput maximization problem (P1) inα (cf. Section VI).
III. DOWNLINKRF ENERGYTRANSFER ANDENERGY
RELAYING OVERRICIANCHANNELS
The energy signals YAR and YAS received respectively at R and S in one slot of ET from A are given by:
YAN = XeApPtA|hAN| e
−ιΘAN+
ℵN,∀ N = {R, S} , (2) where XeA is the zero mean and unit variance energy signal transmitted by A, and PtA is the transmit power of A. |hij| and Θij =
2πd ij
λ − φij
respectively represent the amplitude and phase of the Rician channel fading coefficient for the
link between node i and node j, where i, j = {A, R, S}.
Here 2πdij
λ represents the phase difference due to free space
path delay and φij represents the sum of all other phases that
include phase weights introduced for synchronization, errors due to the local oscillator variations, excess path phase from obstacles, etc. [27]. Lastly, ℵi represents zero mean additive
Using (2) and ignoring EH from noise power [6]–[15], the
received power at R and S in each slot due to RF-ET from
A is given by (3) where† denotes the complex conjugate.
PrAN = (YAN) (YAN) † = PtA|hAN| 2 , ∀ N = {R, S} . (3)
As the received powers PrAS and PrAR at S and R
involve the square of Rician distributed |hAN|, they
fol-low noncentral-χ2 distribution with respective Rice
fac-tors and means as KAS, µPAS =
PtAGAGS (dAS)n 4πλ 2 and KAR, µPAR = PtAGAGR (dAR)n 4πλ 2
. We next discuss the basic probabilistic measures for the received power over Rician channels and then use them for deriving the mean harvested
energy atS due to RF-ET from A, both with and without ER.
A. Basic Properties of Rician Fading Channels
For Rician fading, the channel power gains follow noncentral-χ2distribution with two degrees of freedom. Thus, the probability density function (PDF) fPr of received power Pr,∀x ≥ 0, is given by: fPr(x, K, µP) = e− (K+1)x µP −K µP(K + 1)−1 I0 2 s K(K + 1)x µP ! , (4) where K is Rice factor, µP is mean received power, and Im(·) is the modified Bessel function of the first kind with order m. The Cumulative Distribution Function (CDF) FPr of Pr is:
FPr(x, K, µP) = 1− Q1
√
2K,p2(K + 1)x/µP
, (5)
where Q1(·, ·) is first order Marcum Q-function [28]. Moment
generating function (MGF) ΦPr ofPr can be obtained as: ΦPr(ν, K, µP) (a) = K + 1 K− ινµP + 1 e ιKνµP K−ινµP+1 (b) = K + 1 K− ινµP + 1 e K(K+1) K−ινµP+1−K. (6)
Here (a) is obtained by using [29, eq. (2.17)] and (b) is obtained after a rearrangement in (a).
B. Single-Hop RF-ET from A to S and R
First we obtain mean harvested powerPhAS atS and PhAR
at R in each slot dedicated for RF-ET from A. Using Ph=
L (Pr) as defined in (1) along with PDF fPr and CDF FPr of received power Pr defined in (4) and (5), the PDF fPh of harvested dc powerPh is obtained as:
fPh(x, K, µP) , 1 MjfPr x−C j Mj , K, µP FPr PthN +1 − FPr(Pth1) , (7) where x satisfiesPthj ≤ x−Cj Mj ≤ Pthj+1,∀j ∈ 1, . . . , N. Thus,
using (7), the mean harvested dc powers PhAN whereN =
{R, S} are derived below: PhAN = E [PhAN] = ∞ R 0 x fPhAN x, KAN, µ PAN dx = N P j=1 MjPthj+1+Cj R MjPthj+Cj x(KAN+1)e −(KAN+1)(x−Cj) Mj µPAN −KAN MjµPAN × I0 2 s KAN(KAN+1)(x−Cj) Mj µPAN FPrAN(PthN+1)−FPrAN(Pth1)dx (c) = ∞ P k=0 N P j=1 MjPthj+1+Cj R MjPthj+Cj x e −( KAN+1)(x−Cj) Mj µPAN −KAN MjµPAN k+1 (k!)2 ×(KAN + 1) [KAN(KAN + 1) (x− Cj)] k FPrAN PthN +1 − FPrAN(Pth1) dx = ∞ P k=0 N P j=1 e−KAN(KAN)k [gN ,j(Pthj)−gN ,j(Pthj+1)] (KAN+1)(k!)2[F PrAN(PthN+1)−FPrAN(Pth1)] , (8) where gN ,j Pthj , Cj(KAN + 1) Γ k + 1,(KAN+1)Pthj µ PAN + MjµPNΓ k + 2,(KAN+1)Pthj µ PAN
. Each term in Taylor series expansion of I0(·) used in (c) can be upper bounded as:
KAN(KAN+1)(x−Cj) Mj µPAN !k (k!)2 (d) / (4 KAN(KAN+1)) k (k!)2 (e) ≤ 2 e(KAN+1) k !2k 2πk ≤ 2 e(KAN+1) k 2k . (9)
Here (d) is obtained by knowing that generally (x− Cj) ≤
4MjµPAN, as from (5) Pr (Pr> 4µP) < 0.009,∀K ≥ 1, and (e) is obtained using the Stirling’s approximation [30]: j!≈ √
2π e−jjj+1
2. From (9) we note that the contribution of
hig-her order terms k > ln 1 2WWW000 log(1 ) 4e(KAN+1) −1 is very less than , where 1 and WWW000(x) is the Lambert function
(principal branch) [31]. However, in general for high Rice factor KAN ≥ 10, even considering only first k = e (KAN+1) summands provides a very tight match to the infinite series because the product termx(KAN+1)
MjµPAN e
−KANe−
(KAN+1)(x−Cj)
Mj µPAN
decays very fast with increasing KAN. This has been
nu-merically validated later in Fig. 3 where (8) is shown to be equivalent to the sum of first 30 summands.
So, with T = 1 s as slot duration, mean harvested energy
EhR at R via RF-ET from A over Ns slots is: EhR ,
PhARNs. Similarly for NR and IR (i.e., no ER modes), the mean harvested energy at S is: EnoER
hS ,PhASNs. However for ER and i2ER modes, mean harvested energy at
S via single hop ET fromA over Ns−1 slots is: EhNSs−1 ,PhAS(Ns− 1).
Next we find energy harvested atS, via two-hop ET from R,
in the last slot of RF-ET phase in ER and i2ER modes.
C. Mean Energy Harvested inNsth Slot due to Two-Hop ER
The received energy signal YRS atS in the Nsth slot due
to ER fromR is given by:
YRS = XeR p
βαPtR|hRS| e
−ιΘRS +
where XeR is the zero mean, unit variance energy signal transmitted byR using its energy harvested EhR over Nsslots and PtR is the transmit power ofR. β is indicator variable for relaying with α as fraction of EhRallocated for ER.|hRS| and ΘRS =2πdRS
λ − φRS
respectively represent the amplitude and phase of Rician fading coefficient forR-to-S link. So, we note that if ER takes place from R to S, i.e., α > 0, then S receives two energy signals YASand YRS in the Nsth slot and the received power at S in the Nsth slot is different from (3).
Thus using (3) and (10), the random received power Pr2hopS at S in (Ns)th slot of ER phase, due to vector addition of energy
signals received from A and R, is given by [5, eq. (14)]: P2hop rS = (YAS+ YRS) (YAS + YRS) † =|YAS| 2 +|YRS|2 + 2|YAS| |YRS| e −ι(ΘAS−ΘRS) =PrAS+ βαPrRS + 2pPrASβαPrRSe −ψ2 cos 2π (dAS− dRS) λ , (11) where ψ2is the root mean square phase error term
incorpora-ting the errors due to the local oscillator variations, excess path phase from obstacles, etc. [27]. ψ2 is in radians and e−ψ2
is unit-less. Here vector addition of RF signals of same frequency
received from A and R is considered because these energy
waves can combine constructive or destructively depending on their respective in-phase or out-of-phase addition [5]. So, mean received powerPr2hopS = E
h Pr2hopS
i
, obtained using linearity of expectation and independence ofPrAS andPrRS, is:
Pr2hopS = E [PrAS] + βαE [PrRS] + 2 p βαEhpPrAS i ×EhpPrRS i e−ψ2cos2π (dAS−dRS) λ = µ PAS+βαµPRS + 2pβα µ√ PASµ√PRSe −ψ2 cos 2π (dAS− dRS) λ , (12) where µPRS , (EhR+EiR)GtRGS (dRS)n 4πλ 2 with µPAS = PtAGAGS (dAS)n 4πλ 2
and EhR as defined in Section III-B. Here EiR is the unused harvested energy which is available as the
residual or initial energy at R when NR mode was selected
in previous transmission block. The accumulated energy EiR is zero when any other relaying mode is chosen. µ√
PAS and µ√
PRS in (12) are respectively obtained by substituting µPAS and µPrRS in place of µP in (13) providing E
√ Pr. µ√ Pr , e− K2 2 qπµ P K+1(K + 1) I0 K 2 + K I1 K 2 . (13)
The above expression is obtained by finding the mean of square-root of random variable Pr following noncentral-χ2
distribution with two degrees of freedom.
Using (12) and (13), the mean harvested power atS due to ER in last slot of RF-ET phase is given byPh2hop
S =L
Pr2hopS
. The total energy harvested at S in Ns slots for ER and i2ER
modes is: EER hStot = E
Ns−1
hS +P
2hop
hS . For NR and IR, P
2hop hS = PhAS, which implies that E
noER
hS =PhASNs.
IV. DF RELAY ASSISTEDCOMMUNICATION OVERRICIAN
CHANNELS WITHDIRECTLINK
For the RF-powered IT with T = 1 s, the transmit powers PtR = E
u
hR+EiR and PtS = E
u
hS ofR and S depend on their usable harvested energies EhuR = βh(1− α) PhARNs− P
tx con −Erx con− R0Erxbit i+ and EhuS = hEER hStot − P tx con i+ for IT, as discussed in Sections II-D and III. Here [x]+ = max{0, x}. With PtS as transmit power of S and XiS as zero mean and unit variance information signal, the received signals atR and A due to uplink IT from S in (Ns+ 1)th are:
YSN = XiSpPtS|hSN| e
−ιΘSN +
ℵN, ∀ N = {A, R} . (14) From the received information symbol YSR,R forwards the decoded zero mean, unit power signal dXiS to A using its harvested energy with transmit power PtR in a two-hop half-duplex fashion in the (Ns+ 2) th slot. The RF signal, thus
received atA is given by:
YRA = dXiSpPtR|hRA| e
−ιΘRA +
ℵA. (15)
For Rician fading channel model, the instantaneous signal-to-noise ratio (SNR) γ = Pr
σ2 follows the weighted noncentral-χ
2
distribution with two degrees of freedom. Using (14) and (15), the received SNRs γSR, γRA, and γSA ofto-R, R-to-A, S-to-A links, respectively, are given by:
γN1N2 =PrN1N2 σ2 = PtN1 σ2 hN1N2 2 , (16)
∀ (N1,N2) ={(S, R) , (R, A) , (S, A)} . Due to the
availabi-lity of the direct link and MRC of received signals atA, the received SNR atA involves the sum of γSA and γRA. Next we derive the distribution of this sum to obtain the closed-form expressions for outage probability and normalized throughput atA due to IT from S using harvested energy.
A. Sum of Two Weighted Noncentral-χ2 Random Variables
The distribution of sum of two positive weighted
noncentral-χ2 random variables can be obtained in terms of Laguerre
expansions [32, eq. (3.5)]. However, due to the involvement of complicated recursions in PDF and CDF expressions, its usage has been limited and an integral definition was used in [13]. Here, we present simple expressions for PDF and CDF of this sum by using series expansion of exponential function. The PDF fPr1+Pr2 of sum of two positive weighted
noncentral-χ2random variablesPr1andPr2having respective Rice factor and mean as K1, µP1 and K2, µP2 is given by:
fPr1+Pr2 x, K1, µP1, K2, µP2 = 1 2π Z ∞ −∞ e−ινxΦPr1 ν, K1, µP1 ΦPr2 ν, K2, µP2 dν (f ) = 1 2π ∞ Z −∞ ∞ X j=0 ∞ X k=0 K1(K1+1) K1−ινµP1+1 j+1 K2(K2+1) K2−ινµP2+1 k+1 dν j! k! K1K2eK1+K2+ινx (g) = ∞ X j=0 ∞ X k=0 K1(K1+ 1) µP1 !j+1 K2(K2+ 1) µP2 !k+1 xj+k+1
× 1F1 j + 1; j + k + 2; K2+1 µ P2 − K1+1 µ P1 x K1K2j! k! Γ(j + k + 2) e K1+(K2+1)xµ P2 +K2 , (17)
where (f ) is obtained by substituting series expansion of ex=
∞
X
k=0
xk
k! in (6). (g) is obtained by using a readily available result given by [28, eq. (3.384.7)] in (f ). 1F1(a; b; x) =
Γ(b) 1 R 0 extta−1(1−t)b−a−1dt Γ(b−a)Γ(a) = ∞ X k=0 (a)kxk
(b)kk! is the confluent hy-pergeometric function of the first kind [28], and (a)k, (b)k are the Pochhammer symbols defined as: (a)k= Γ(a+k)Γ(a) . Γ (x) =
∞
R
0
tx−1e−tdt is the gamma function. As e−Ki e Ki(Ki+1) Ki−ινµPi+1 < 1∀i = {1, 2}, e Ki(Ki+1) Ki−ινµPi+1 −Ki and e−Kij! Ki(Ki+1) Ki−ινµPi+1 j
are integrable, and we can swap the integration and double summation [30, Theorem 16.7] in (f) to obtain the result
given by (17). Further, the Lagrange remainder Rj in
j-term finite Taylor series approximation of exponential function e Ki(Ki+1) Ki−ινµPi+1 −Ki with z∈ 0, Ki(Ki+1) Ki−ινµPi+1 is given by: |Rj| = e−Ki ez (j+1)! Ki(Ki+1) Ki−ινµPi+1 j+1 (h1) < (Ki)j+1 (j+1)! (h2) ≤ (e Ki)j+1 √ 2π(j+1)j+1+ 12 (h3) < e Ki j+1 j+1 , (18)
where (h1) is obtained by using z < Ki(Ki+1)
Ki−ινµPi+1 and νµPi 2 > 0; (h2) is obtained using the Stirling’s approximation [30]: j!≈√2π e−jjj+1
2; and (h3) holds as√2πj > 1. Using (18), the minimum numbers of terms j∗to ensure that the Lagrange remainder Rj∗ after considering j∗ terms is less than , i.e.,
|Rj∗| < , is given by: j∗ ≥ log 1
W0 WW00 log(1 ) eKi −1 − 1. To gain further insight, we note that for high Rice factor values Ki≥ 10, j∗≈ eKi= 2.72 Ki.
PDF ofPr1+Pr2 for Rayleigh fading case (K1= K2= 0) can be obtained in simple form from (17) by considering single term j = k = 0 in double summation with K1, K2→ 0:
fRay Pr1 +Pr2 x, µP1, µP2 = e− x µP1 − e− x µP2 µP1− µP2 ,∀x ≥ 0. (19)
The CDF of Pr1+Pr2, obtained using series representation of 1F1(a; b; x) = ∞ P i=0 (a)ixi (b)ii! in (17), is given by: FPr1+Pr2 x, K1, µP1, K2, µP2 = ∞ X i=0 ∞ X j=0 ∞ X k=0 K1jKk 2(j + 1)i eK1+K2 × Kµ1+ 1 P1 !j+1 K2+ 1 µP2 !−i−j−1 K2+ 1 µP2 − K1+ 1 µP1 !i × Γ(i + j + k + 2)− Γ i + j + k + 2,(K2+1)x µP2 (j + k + 2)ii! j! k! Γ(j + k + 2) . (20) Γ (a, x) = R∞ x t
a−1e−tdt is upper incomplete gamma
function. As with a = j + 1 < j + k + 2 = b, (a)ix i
(b)ii! ≤ xi
i!,
we note that j∗ for the Lagrange remainder Rj∗ in the Taylor series expansion of1F1(a; b; x) to be less than , is lesser than
in case of Lagrange remainder in the Taylor series expansion of exponential function e (x). Hence, although (20) involves three series, each of the three infinite sum-terms converge very quickly. Also we show later in Fig. 4 that practically this CDF reduces to a finite sum with only 30 summands in each series. Similarly using (19), the expression for CDF ofPr1+Pr2 for Rayleigh fading channels is:
FRay Pr1 +Pr2 x, µP1, µP2 = 1 − µP1e− x µP1 − µP2e − x µP2 µP1− µP2 . (21)
Using (19) and (21), the i2ER performance over Rayleigh
fading channels can be investigated.
B. Outage Analysis for RF-powered DF-IR with MRC The outage probability pout is the probability that the data
rate received atA during IT (and IR) phase (of duration 1 or 2 slots depending on relaying mode) falls below a spectral efficiency threshold R0 bits/sec/Hz or bps/Hz. Considering
half-duplex DF-IT fromS to A via R with MRC at A due to
direct link availability, the outage probability pIR
out for IR or
i2ER over the Rician channels is given by:
pIRout (i1) = Pr 1 2log2(1 + min{γSR, γRA+ γSA}) < R0 = Pr min{γSR, γRA+ γSA} < 2 2R0− 1 = 1− Pr γSR> 2 2R0 − 1 Pr γRA+ γSA > 2 2R0 − 1 (i2) = 1−h1− FPrSR 22R0 − 1, KSR, µPSR σ2 i h 1− FPrRA+PrSA 22R0− 1, K RA, µ PRA σ2 , KSA, µ PSA σ2 i , (22) where (i1) is due to half-duplex DF-IR with MRC [13]. (i2) is obtained using (16) and fact that γN1,N2 follows noncentral-χ2 distribution with two degrees of freedom, Rice factor KN1N2, and mean µ PN1 N2 σ2 = PtN 1GN1GN2 dN1N2nσ2 λ 4π 2 , ∀ (N1,N2) = {(S, R) , (R, A) , (S, A)}. So pIR
out in (22) can be obtained
in closed-form by using CDFs defined in (5) and (20) with appropriate arguments as mentioned in (22). Also, using (21) along with K = 0 in (5) and (22), the outage probability pIR,Rayout for IR over Rayleigh fading channels is given by:
pIR,Rayout = 1− 1 µPRA− µPSA µ PRAe −(22R0−1) σ2 µ PRA+ σ2 µ PSR − µPSAe −(22R0−1) σ2 µ PSA+ σ2 µ PSR . (23)
Further, for no IR cases, i.e., NR and ER modes, pout = pnoIRout,
obtained using (5), is defined below: pnoIR out = 1− Q1 p2KSA, r 2(KSA+1)(2R0−1)σ2 µ PSA . (24)
Start Decision II One-hop IT Two-hop IR NR mode ER mode IR mode i2ER mode (ER or not?) Decision II (ER or not?) Decision I
(IR or not?) Yes
Yes
Yes No
No
No
Fig. 2:Decision tree for optimal relaying mode selection policy.
C. Achievable Normalized Throughput in RF-powered i2ER
Normalized throughput τ is the amount of successfully transmitted data per unit time in each communication slot. Considering a delay-limited scenario with rate constraint
(outage threshold) of R0 bps/Hz at A for IT phase of 1 or
2 slots (Table I), τ for different relaying modes is: τ , R0(1−pnoIRout) Ns+1 , NR (β = 0) and ER (β = α = 1) 2R0(1−pIRout) Ns+2 , IR (β = 1, α = 0) and i 2ER (β = 1).(25)
V. OPTIMALMODESELECTION ATRF-EH RELAY
We now discuss the insights on which mode to choose
among NR, ER, IR, and i2ER. This optimal mode selection
policy for efficient outage performance basically involves two main decision making: (i) Two-hop IT with IR or single-hop
IT without IR (Section V-B), and (ii) RF-powered S-to-A IT
with ER or without ER (Section V-A). Fig. 2 summarizes the decision making process in the optimal relaying mode selection policy. Subsequently, we derive conditions for the improved performance of cooperative ER, IR, and i2ER modes over non-cooperative NR mode.
A. Feasibility of Energy Relaying (ER) Mode
First we derive conditions for improved performance of ER
over NR. ER mode is useful when the transmit power PtS
of S, based on its harvested energy EER
hStot from A and R
jointly, is more than the harvested energy EnoER
hS fromA alone,
i.e., without ER. Knowing EnoER
hS = PhASNs and PhAS ≈ L µPAS, E
ER
hStot defined in Section III-C can be rewritten as:
EERh Stot = E Ns−1 hS +P 2hop hS =PhAS(Ns− 1) + Lµ PAS + βαµPRS + µ0 q βαµ PASµPRS , (26) where µ0, cos 2π(dAS−dRS) λ (K + 1) I0 K2 +K I1 K2 2 ×πe−K−ψ2
2(K+1) . Using (26) and discussion in Sections III-C
and IV-B, PtS using harvested energy E
ER
hStot over Ns slots is:
PtS = E ER hStot−P tx con=PhAS(Ns−1)+Mj µ PAS + βαµPRS + µ0 q βαµPASµPRS+Cj− Pcontx ,∀j ∈ n (1≤ j ≤ N) ∧ Pthj≤µPAS+βαµPRS+µ0 q βαµPASµPRS≤Pthj+1 o .(27) Next we present an important result on utility of ER mode based on the variation of α.
Lemma 1: With EhER Stot > P
tx
con, the transmit power PtS of
S based on its harvested energy EER
hStot via RF-ET from A
and ER fromR is either: (i) concave increasing function of α when energy signals received atS in the Nsth slot fromA
andR add up constructively, or (ii) strictly-convex in α when energy signals fromA and R lead to destructive interference. Proof:First of all we note that, the value of µ0containing
cosineterm cos (·) represents constructive or destructive inter-ference of energy signals fromA and R. µ0> 0 always leads
to constructive interference, i.e., EER
hStot > EhnoERStot. However if
µ0 < 0, then received energy signals at S in Nsth slot may
add up destructively to cause EhER Stot ≤ E noER hStot. As ∂ 2P tS ∂α2 = − µ0Mj q αβµ PASµPRS
4α2 , we can observe that
PtS is concave in α if µ0 ≥ 0; otherwise it is a
con-vex function of α ∀µ0 < 0. We also note that, since
∂PtS ∂α = 1 2βµPRSMj µ0µPAS q αβµ PASµPRS +2, PtS is strictly
increasing function of α ∀µ0 ≥ 0 ∧ β = 1. On other
hand, if µ0 < 0, a unique feasible critical point αgER = n α ∂P tS ∂α = 0 ∧ (0 ≤ α ≤ 1)o is defined as: αgER , µ20µ PAS 4βµ PRS
. Thus, for µ0 < 0 if α ≤ αgER, then PtS is a decreasing function of α. However when 4αgER < α ≤ 1, PtS is an increasing function of α and even for µ0 < 0, Lµ PAS + βαµPRS + µ0 q βαµ PASµPRS >PhAS, which leads to the improved ER performance over NR.
Remark 1: Though µ0< 0 leads to destructive interference
of signals fromA and R, i.e., LµPAS+βαµPRS≥LµPAS +βαµPRS+µ0 q βαµPASµPRS,EERh Stot> E noER hStot, ifα > 4αgER. Remark 2: ER is always beneficial over NR∀α > 0 if µ0≥
0 and∀α > 4αgER ifµ0< 0.
Hence, we conclude that ER is preferred over NR when
energy signals received at S from A and R are combined
constructively. The chances of ER performing better than NR increase with increasing PtR which leads to a higher µPRS because it helps in meeting the condition α > 4αgER.
B. Feasibility of Information Relaying (IR) Mode
Now we obtain the feasibility conditions for improved performance of IR over NR mode.
Lemma 2:The normalized delay-limited throughput τ in IR is more than NR if and only if one of these two conditions hold: (i) pIR
out≤ pnoIRout or (ii) pnoIRout < pIRout <
(Ns+2)pnoIRout+Ns 2(Ns+1) . Proof: Firstly, τNR τIR = (Ns+2)(1−pnoIRout) 2(Ns+1)(1−pIRout) and 12< Ns+2 2(Ns+1)< 1,∀Ns> 0. For τIR> τNR, we require pnoIRout > pIRout−
Ns(1−pIRout)
Ns+2 , which is true∀pIRout≤ pnoIRout. Thus, for τIR> τNR, either pIRout≤
pnoIR
out or pnoIRout < pIRout <
(Ns+2)pnoIRout+Ns 2(Ns+1) . As p noIR out = Pr γSA < 2R0−1 and pIR out= Pr min{γSR, γRA+ γSA} < 2 2R0− 1, it is worth noting that the outage threshold of 2R0 for IR is
two times the outage threshold R0 for NR.
Following this result, we next discuss the conditions where outage probability pIRout in IR and i2ER modes is better than
Lemma 3:To ensure that pIRout< pnoIRout, the following average
SNR conditions should be met: (i) E [γSR] > E [γSA] E [γSA] +2 and (ii) E [γRA] > E [γSA] (1 + E [γSA]).
Proof: Please refer to Appendix A
Remark 3: With E [γSA] > 0, Lemma 3 implies that for feasibility of IR, E [γSR] > E [γRA]. Or, IR is feasible when R is placed relatively closer to S to strengthen S-to-R link.
Corollary 1: Outage performance for IR is better than NR if relay placement (RP) (dSR, dRA) lies in the set SIR , (dSR, dRA) dSR < dSA GR GD (2+E[γSA])n1 ∧ dRA < d max RA .
Proof: From Lemma 3, we note that to ensure
pIRout < pnoIRout, two average SNR conditions (i) and (ii)
should be met. Condition (i) on simplification results in the following relationship betweenS-to-R and S-to-A distances:
dSR < dSAGRGD 2+(NsPhAS−P txcon)GAGS λ2 (4πσ)2(dSA)n !n1 = dSA GR GD (2+E[γSA])n1.
Similarly, condition (ii) puts an upper bound dmaxRA on
R-to-A distance dRA to meet the EH requirements of
R for efficient IR, which is given by: dRA < d
max RA , dRA 16π2E[γSA](1+E[γSA])σ2(dRA)n
βGAGRλ2(PhARNs(1−α)+EiR−Pcontx−Erxcon−R0Ebitrx) = 1
. These bounds on dSR and dRA form the feasible RP set SIR for enhanced outage performance of IR over NR.
C. Insights on Optimal Mode Selection Policy
Using the observations in Sections V-A and V-B, now we provide insights on the mode to be selected among NR, ER, IR, and i2ER based on the two decision making (cf. Fig. 2) for minimizing outage probability. From Lemma 2, improved outage performance in IR or i2ER mode also implies that their throughput performance is better than ER or NR mode.
1) Decision I: IR or no IR?: From Lemma 3 and
Corol-lary 1, we note that if relay placement (dSR, dRA) ∈ SIR, then the outage probability pIR
out in two-hop IT in IR and
i2ER modes is better or lower than the outage probability pnoIR
out in single-hop IT in ER and NR modes. So with the
available statistical CSI, decoding capability of R based on its harvested energy andS-to-R link quality is decided. Only when decoding capability is sufficiently large such that either
of IR or i2ER modes perform better than ER or NR modes,
R invests its harvested energy on IR. Otherwise, it utilizes its energy for ER or saves it for future if a NR is chosen.
Further, as i2ER with α = 0 reduces to IR mode, the
feasibility conditions for i2ER mode are similar to as in IR
mode, which are mentioned in Lemma 3. However, when both IR and i2ER are feasible, i.e., pIR
out < pnoIRout for α = 0,
then i2ER can provide better performance than IR by allowing
integrated IR and ER, as discussed next.
2) Decision II: ER or no ER?: The decision for ER depends
on whether the received energy signals from A and R add
constructively or destructively. The conditions for preferring
ER over NR mode based on the value of α and µ0have been
presented in Lemma 1 and Remark 2.
When IR mode is feasible, then for µ0 ≥ 0, i2ER can
provide better outage performance if ∂E[γRA+γSA]
∂α ≥ 0. In
other words, if both E [γSR] and E [γRA + γSA] are increasing
in α, then from Theorem 1 (Section VI-B), pIRout in i2ER is a
decreasing function of α implying that its outage performance with α > 0 is better than that of IR mode having α = 0.
VI. OPTIMALSHARING OFHARVESTEDENERGY ATR
Following the observations in previous section, now we optimize α to efficiently utilize the available harvested energy
at R for ER and IR. First we formulate the optimization
problem, followed by its generalized-convexity proof and the joint global-optimal solution (R0∗, α∗, β∗).
A. Optimization Formulation
We intend to maximize the normalized throughput τ by efficiently dividing harvested energy at R over Ns slots, i.e.,
α fraction for ER and remaining (1− α) fraction for IR. As τ defined in (25) is a function of rate constraint R0 bps/Hz
to be met at A during the IT phase of 1 or 2 slots, we also
find maximum achievable rate R0 that can be met with high
probability 1− pthout, where pthout 1. This is denoted by
constraint C1 in throughput maximization problem (P1). (P1) : maximize
R0, α, β
τ , subject to: C1 : pout ≤ pthout,
C2 : α≥ 0, C3 : α ≤ 1, C4 : β ∈ {0, 1} . (28)
Here β = 1 or β = 0 is respectively based on whether to go for relaying (ER, IR, or i2ER) or not (NR). As (P1) is nonconvex, it is difficult to jointly optimize R0, α, and β. So,
we break the problem (P1) into two parts, i.e., first solve outage minimization problem (P2) to find optimal α that minimizes pIRout. After that we use monotonicity of pout in
R0 to iteratively solve (P1).
(P2) : minimize
α p
IR
out, subject to: C2 and C3. (29)
Remark 4: Using the statistical CSI along with the system parameters mentioned in Sections II, III, and IV, energy-rich A solves (P1) and informs R and S respectively about the optimal relaying mode (α∗, β∗) and optimal R∗0 to maximize
the normalized delay-limited throughput.
B. Generalized-Convexity of Outage Minimization Problem Here we present some important results in the form of Lemma 4, Corollary 2, and Theorem 1, that will be useful in proving conditional generalized-convexity [33] of (P2).
Lemma 4:The average SNR E [γRA+ γSA] forS-R-A link is: (a) strictly concave in α if µ0> 0 and (b) convex function
of α for µ0≤ 0 with unique stationary point denoted by αgIR. Proof:Using linearity of expectation in E [γRA+ γSA], E h γRA+ γSA i = E [γRA] + E [γSA] = µPSR σ2 + µPSA σ2 =
(1−α)PhARNs+EiR−Pcontx−E rx con−R0Ebitrx (β GR)−1(dRA) n +PhAS(Ns−1)+MjP 2hop hS+Cj−P tx con (GS)−1(dSA)n GA σ2 λ 4π 2 , ∀j ∈n(1≤ j ≤ N) ∧Pthj≤P 2hop hS ≤Pthj+1 o , (30)
wherePh2hopS = µ PAS + βαµPRS+ µ0 q βαµ PASµPRS. From ∂2E[γRA+γSA] ∂α2 =− GSGAλ2µ0Mj q αβµ PsµPRS 4(4π)2σ2α2(d SA) n , we note that depending on whether µ0 > 0 or µ0 ≤ 0, E [γRA+ γSA] is respectively strictly concave or convex in α. Further, it may also be noted that ∂E[γRA+γSA]
∂α = GAλ2 (4πσ)2 µ0µPs q αβµ PsµPRS + 2 βG Sµ PRSMj 2(dSA)n − βGRNsPhAR (dRA)n
, using which the unique gradient point αgIR, satisfying
∂E[γRA+γSA] ∂α = 0, is given by: αgIR = 1 4β (d RA) n GSµ0√µPASµPRSMj (dSA)n GRNsPhAR−(dRA) n GSµ PRSMj 2 . So for µ0 ≤ 0, E [γRA+ γSA] is strictly decreasing in α∈ (0, αgIR) and strictly increasing in α∈ (αgIR, 1).
Corollary 2: The average SNR E [γSR] for S-to-R link is concave increasing in α if µ0> 0, and strictly-convex function
of α for µ0≤ 0 with unique critical point αgER, if it exists.
Proof: As defined in Section IV-B, the average SNR
E [γSR] forS-to-R link is given by:
E [γSR] = µPSR σ2 = PtSGSGR σ2(d RS) n λ 4π 2 (31) where PtS is defined in (27). As E [γSR] is positive affine function of PtS, from Lemma 1 we observe that E [γSR] is respectively concave increasing and strictly-convex in α for µ0> 0 and µ0≤ 0. The unique critical point αgER =
µ2 0µPAS
4βµ
PRS for µ0≤ 0, if exists, is defined in Lemma 1.
Theorem 1: The complimentary CDFs (CCDFs) FPrSR
and FPrRA+PrSA of received powers PSR andPRA+PSA are respectively positive increasing log-concave functions of E [γSR] and E [γRA+ γSA].
Proof: Please refer to Appendix B.
By using these results, the conditional-pseudoconvexity and global-optimality of (P2) are discussed next.
Theorem 2:As the objective function of (P2) is
pseudocon-vex in α∈
(
[0 , 1] , µ0> 0
[αgER, αgIR] , µ0≤ 0,
and constraints C2, C3 are affine functions of α, there exists a unique global-optimal solution α∗∈ [αgER, αgIR] that minimizes p
IR out.
Proof: Please refer to Appendix C.
C. Global Optimal Allocation of Harvested Energy at Relay
Theorem 3: The global optimal utilization of
harves-ted energy at R for minimizing pIR
out is given by α∗ , argmin α={0,αout,1} pIRout α=0, p IR out α=α out , pnoIRout α=1 , with αout,
αoutmin, αcriout< αoutmin, αout
cri, αminout ≤ αoutcri ≤ αoutmax,
αoutmax, otherwise.
(32)
Here αminout, maxn
z+µ20µ PASMj+ r µ2 0µPAS 2z+µ2 0µPASMj 2βµ PRSMj , 0, αgER o , αoutcri,nα ∂pIR out ∂α = 0
∧ (0<α< 1)o, αoutmax, minn1,
αgIR, 1− ErxbitR0+Econrx+P tx con−EiR NsPhAR o , and z,−2 Cj+µPASMj+ (Ns−1)PhAS−P tx con
+µ20µPASMj withPh2hopS ∈Pthj,Pthj+1. Proof: As pIR
out is pseudoconvex or unimodal1 in α ∈
[αgER, αgIR]∀µ0 (cf. Theorem 2), the global-optimal solution
α∗ is given by the unique mode αout
cri, defined in (C.3)
by solving ∂pIRout
∂α = 0, if it exists in the feasible region
defined by C2–C3 for µ0 > 0 or satisfies condition αoutcri ∈
[αgER, αgIR] ,∀µ0 ≤ 0. However if α out cri > α out max or α out cri <
αoutmin, then due to the corresponding monotonically decreasing or increasing trend of pIR
out with α, global-optimal α∗ is
given by the two corner points αoutmax and αoutmin, respectively.
αoutmax ensures that C3 is satisfied, αout ≤ αgIR∀µ0 ≤ 0 and PtR > 0. Similarly, α
out
min ensures that C2 is satisfied,
αout ≥ αgER∀µ0 ≤ 0 and PtS > 0. As for µ0 ≤ 0, p
IR out
respectively follows monotonically increasing and decreasing trend with α, for α < αgER and α > αgIR, optimal α
∗ is given
by one of the three potential candidates, i.e., α∈ {0, αout, 1}.
Also, it may be noted that forµ0> 0, αout= αoutcri itself.
Remark 5: Althoughα = 1 is shown as a feasible solution for (P2) to minimize pIR
out,α = 1 leads to PtR = 0 implying
that no communication takes place from R-to-A, i.e. no IR
for α = 1. So if α∗= 1, then pout= pnoIRout as given by(24).
D. Iterative Scheme to Maximize Normalized Throughput τ
Now we try to maximize τ by jointly optimizing R0, α, and
β in problem (P1). Since (P1) is nonconvex and has combi-natorial aspect due to inherent mode selection in definition of τ given in (25), we make use of Theorem 3 to find optimal α∗ for a given R
0 with β = 1. In this regard we present an
iterative scheme, named Algorithm 1, that helps in maximizing τ by iteratively optimizing α and R0. Here while optimizing
R0 to meet certain quality-of-service (QoS) requirement, we
need to ensure that corresponding pout < pthout(constraint C1).
The iterative scheme starts with finding R0 that satisfies
pnoIR
out ≤ pthout for ER and NR modes, denoted by RER0 and
RNR
0 , respectively. These RER0 and RNR0 values are obtained by
finding inverse of the CDF function ofPrSAusing Algorithm 2 with α = β = 1 and α = β = 0, respectively. With initial
R0 being R (0) 0 = Ns+2 2(Ns+1)max{R ER 0 , RNR0 }, we find optimal α∗minimizing pIR
out by using Theorem 3. If 0 < α∗< 1 with
β(1) = 1, then this implies that neither of NR or ER modes can provide the optimal R0∗, or in other words, R∗0 > R
(0) 0 .
So α∗ = α(1) with R
0 = R (0)
0 results in a p∗out which is
lower than pth
out and the Algorithm 1 continues. This decrease
in p∗out implies that we can achieve higher R0 due to the
improved end-to-end SNR quality. Next we find updated R0,
denoted by R(1)0 , satisfying pIR
out ≤ pthout for i2ER by using
inverse of CDF of min{PrSR,PrRA+PrSA} with E [PrSR] and E [PrRA+PrSA] defined using α = α
∗. The iterative
process continues till R (i) 0 − R (i−1) 0 ≤ξR0, where ξR0 1 is the acceptable tolerance. Algorithm 1 terminates with the optimal R∗0, α∗, β∗ that provide maximum τ∗ by selecting
the optimal relaying mode and maximum achievable rate R∗0
1It may be noted that unimodality (having unique minima) of a single
Algorithm 1 Iterative scheme to maximize normalized throughput τ by jointly optimizing R0, α, β
Input: Relay position (dRA, dSR), system and channel parameters
(cf. Section II), with tolerances pthout, ξR0, ξ
Output: Maximized throughput τ∗
along with optimal R∗0, α ∗
, β∗ (A) Initialization
1: Call Algorithm 2 to find ΥER=
n Υ p th out− FPrSA Υ, KSA, µ PSA σ2 ≤ ξ o
for S-to-A link in ER mode with Pr =
PrSA, α = 1, β = 1, andpdout= p
th out 2: Call Algorithm 2 to find ΥNR=
n Υ p th out−FPrSA Υ, KSA, µ PSA σ2 ≤ ξ o
for S-to-A link in NR mode with Pr =
PrSA, α = 0, β = 0, andpdout= p
th out 3: Set i ← 0, R(0)0 ← Ns+2
2(Ns+1)log2(1 + max {ΥER, ΥNR})
(B) Recursion
4: repeat (Main Loop)
5: Set i ← i + 1, α∗(i) ← α ∗
that minimizes pIR out for
achieving rate R(i−1)0 in i2ER using Theorem 3 6: Call Algorithm 2 to find Υ(i)SRA = nΥ
p th out− 1 −h1− FPrSR Υ, KSR,µPSRσ2 i h 1 − FPrRA+PrSA Υ, KRA,µPRAσ2 , KSA, µ PSA σ2 i ≤ ξo in i2ER with Pr = min{PrSR, PrRA+ PrSA}, α = α ∗ (i), β = 1,pdout= p th out 7: Set R(i)0 ← 1 2log2 1 + Υ(i)SRA 8: until R (i) 0 − R (i−1) 0 ≤ ξR0
(C) Termination with Optimal Solution
9: Set R0,1= log2(1+ΥER), R0,2= log2(1+ΥNR), R0,3= R(i)0 10: Set j∗ ← argmax
1≤j≤3 R
0,j, and optimal {R∗0, α∗, β∗} is given
by {R0,1, 1, 1} , j∗= 1 (ER mode) {R0,2, 0, 0} , j∗= 2 (NR mode) R0,3, α∗(i), 1 , j ∗
= 3 (i2ER mode (IR if α∗(i)= 0))
with pout ≤ pthout. Thus, with increasing iteration (i), {R (i) 0 }
monotonically increases (i.e., R(i+1)0 > R(i)0 ) because of monotonically improving end-to-end SNR due to the optimal relaying mode selection for increasing {R(i)0 }.
Fast Convergence of Algorithms 1 and 2: Due to strict
monotonicity and pseudoconvexity of pout in R0 and α
respectively, α∗(i) in each iteration can be found efficiently and in general Algorithm 1 converges to acceptable optimal solution R∗0 in two to three iterations only.
Similarly, Algorithm 2 employing a modified version of Newton-Raphson method, provides fast convergence to the inverse Υ of CDF FPr, where Υ is defined in steps 1, 2, and 6 of Algorithm 1 for Pr in different relaying modes, due to the
following properties: (i) FPr is monotonically increasing in Υ. (ii) FPr is continuously differentiable positive log-concave in Υ ∈ [0, ∞). (iii) ∂FPr
∂Υ is continuously differentiable
log-concave function of Υ. (iv) EPr
σ2
provides a very good starting point. We noted that with conventional update equation Υ(i)
←Υ(i−1)+ F(i−1)−[1−pdout]
fPrSA(Υ(i−1),K SA,
µSA σ2 )
in standard Newton-Raphson method, iterations sometimes diverge. To overcome this drawback we consider the usage of log function with which convergence improves significantly. Via extensive
nu-Algorithm 2Iterative scheme to obtain inverse Υ = FPr −1
(pdout)
satisfyingpdout= FPr(Υ, K, µP)
Input: CDF FPr, PDF fPr, and mean µP
σ2 of Pr
σ2 along with α, β,
K, and tolerance ξ for acceptable outage probabilitypdout. Here
Pr ∈ {PrSA, min {PrSR, PrRA+ PrSA}}. Output: Inverse Υ∗ = ( Υ d pout− FPr Υ, K, µP σ2 ≤ ξ ) of FPr 1: Set i ← 0 2: if (Pr= PrSA)then 3: Set Υ(0)←µPSA σ2 , F (0) ← 1 − FPrSA Υ(0), KSA, µ PSA σ2
4: else if (Pr= min {PrSR, PrRA+ PrSA})then 5: Set Υ(0)← min n µ PSR, µPSA+µPRA o σ2 6: Set Fa (0) ← 1 − FPrSR Υ(0), KSR, µ PSR σ2 , 7: Set Fb (0) ← 1−FPrRA+PrSA Υ(0), KRA, µ PRA σ2 , KSA, µ PSA σ2 8: Set F(0)← Fa (0) · Fb (0) 9: repeat (Main Loop) 10: Set i ← i + 1
11: if (Pr= PrSA)then 12: Set Υ(i)← Υ(i−1)
+F
(i−1)h log2
h
F(i−1)i−log2[1−pout]d i fPrSA Υ(i−1),KSA, µ PSA σ2
13: Set F(i)← 1 − FPrSA
Υ(i), K SA, µ PSA σ2
14: else if (Pr= min {PrSR, PrRA+ PrSA})then 15: Set Υ(i) ← Υ(i−1) + hlog
2
h
F(i−1)i− log2[1−pdout]
i ×F(i−1) Fb (i−1) fPrSR Υ(i−1), K SR, µ PSR σ2 + Fa (i−1) ×fPrRA+PrSA Υ(i−1), KRA, µ PRA σ2 , KSA, µ PSA σ2 −1 16: Set Fa (i) ← 1 − FPrSR Υ(i), KSR, µ PSR σ2 , Fb (i) ← 1 − FPrRA+PrSA Υ(i), K RA, µ PRA σ2 , KSA, µ PSA σ2 17: Set F(i)← Fa (i) · Fb (i) 18: until dpout− 1 − F(i) ≤ ξ
merical results, we have found that on an average Algorithm 2 converges to acceptable tolerance ξ in less than 20 iterations. E. Some Additional Insights on Key System Parameters
1) DecidingNsSlots Dedicated for RF-ET: Since the
end-to-end ET efficiency is very low, we need to allocate sufficient time for ET so that both S and R have sufficient harvested energy to carry out uplink IT at a desirably rate R0. The rate
of change of τ for NR mode with Ns is:
∂τ ∂Ns = R0 (Ns+1)2 (2R0−1)fPrSA 2R0−1,KSA, µ PSA σ2 NsµPSA h (Ns+1) µ PSA+Pcontx i−1 −1− FPrSA 2R0− 1, K SA, µ PSA σ2 . (33)
From (33) we note that for low values of R0, ∂N∂τs < 0,
implying that τ in NR is a decreasing function of Nsbecause
for low R0, PDF fPrSA is lower than CCDF
h
1−FPrSA
i and thus Ns can be set as the minimum, i.e., 2 slots. However if
R0is high for meeting the demands of high QoS applications,
then τ initially increases till Ns = Ns∗ and for Ns > Ns∗ it
decreases with increased Ns. Here the optimal Ns for NR,
2) Insights on Optimal Relay Placement (ORP): Although this work focuses on solving the dilemma ofR on whether to
cooperate in downlink ER to S or uplink IR to A based on
its relative placement betweenA and S, here we give insights on ORP for different relaying modes.
For ER mode, detailed investigation on the ORP in two-hop RF-ET was carried out in [5]. It was observed that ORP, always lying in the constructive interference region, depends on the end-to-end distance dAS. If dAS is relatively low then ORP lies in the constructive interference region closer to the RF-EH deviceS, whereas if dAS is relatively high then ORP lies closer to RF source A. We have obtained similar results as plotted in Fig. 6 and discussed in Section VII-B.
Regarding IR mode, it is difficult to obtain the closed-form results for ORP due to high composite non-linearity. However by exploiting the behavior of DF-IR protocol, we provide a suboptimal RP solution that provides tight approximation to the global-ORP. As the DF-IR performance is bottlenecked by
the minimum of the SNR of S-to-R link and the SNR due
to MRC, we present a suboptimal RP that improves the SNR of the bottleneck link by making the two SNRs equal. This RP solution is obtained by solving E [γSR] = E [γSA+ γRA]. Further as E [γSA] > 0,R is placed closer to S to ensure that E [γSR] > E [γRA] The goodness of this suboptimal solution providing insights on the features of the global optimal RP solution is investigated numerically in Section VII-B.
Finally with the above discussions on ORP in ER and IR, we note that the ORP in i2ER not only lies closer to S to ensure efficient IR but also it should fall in the constructive interference region to ensure efficient ER. This claim is also numerically validated later in Section VII-C.
VII. NUMERICALRESULTS ANDDISCUSSION
We conduct numerical investigation on performance of WPCN under different relaying options: NR, ER, IR, or i2ER.
Unless otherwise stated, the considered system parameters are: PtA = 30 dBm, GA= GS= 1 dBi, GR= 6.1 dBi, σ 2= −100 dBm, λ = 0.328 m, yR0={0.25, 0.05} m [5] for dAS ={1, 2} m, n = 2, T = 1 s, ψ2= 0.175 rad [5], Ptx con= 0 W, Econrx = 0.927 mJ [26], Erx
bit= 93.53 µJ [26], EiR = 0 J, K = 10 dB for all the links, and tolerances as ξR0 = 10
−3, ξ = 10−6.
Using (1), the piecewise linear approximationPh=L (Pr)
for Ph (in mW) at the output of the commercially available
Powercast P1110 RF harvester [24] can be obtained with Pth = {0.282, 0.501, 1.0, 3.548, 25.119, 100} mW as six
re-ceived threshold powers dividing the harvested-rere-ceived power characteristic of P1110 into N = 5 linear pieces having slope M = {0.857, 0.786, 0.485, 0.733, 0.465} and intercept C = {−0.223, −0.194, 0.107, −0.772, 5.948} mW.
The accuracy of approximation (1) can be observed from the fact that root mean square error (RMSE) in approximating the measured results given in [5, Fig. 5(b)] is less than 0.0003 and corresponding R-square statistics value is more than 0.9997. A. Validation of Analysis
First, we validate the analytical expression for EhER
Stot derived using (8) and (12). Analytical results in Fig. 3 are generated
Position (x-coordinate) of relay (m)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 M ea n h ar ve st ed en er gy at in fo rm at io n so u rc e S (m J ) 5 10
15 Analysis Simulation No energy relaying Ns= 20
Ns= 5
Ns= 15 Ns= 10
Fig. 3:Variation of EhERStot with relay position (xR, 0.25 m) and Ns.
x 0 5 10 15 20 25 C D F of su m FP r 1 + Pr 2 (x ) 0 0.25 0.5 0.75 1 Analysis Simulation 13.4 13.5 13.6 0.876 0.88 0.884 0.888 K= 5, µP1= 3, µP2= 6 K= 2, µP1= 2, µP2= 4 K= 1, µP1= 1, µP2= 2
Fig. 4: Validation of expression (20) for CDF of sum of two weighted noncentral-χ2 random variables.
using only first 30 summands of series in (8). The simulation results on mean harvested power atS for varying relay position (xR, 0.25 m) and Ns are generated by finding mean of 10
7
random realizations of harvested dc power PhAS obtained
by applying (1) on random received power PrAS following
noncentral-χ2 distribution. A close match between analytical and simulation results as observed in Fig. 3 validates the analysis in Section III with a RMSE of less than10−4.From Fig. 3 it is observed that, in comparison to energy harvested EnoER
hS in no ER case, E
ER
hStot in ER is affected by constructive
and destructive interference of energy signals received from A and R. However with increasing Ns, the destructive
inter-ference region decreases due to increasedEhR, which results
in improved ER gain with higher RF-ET fromR.
Next we validate the outage analysis carried out in Section IV. We have considered only first 30 summands for each of the three series in (20) for generating analytical results depicted by solid line in Fig. 4 and different line styles in Fig. 5. We first validate expression (20) for CDF of sum of two weighted noncentral-χ2 random variables in Fig. 4 for different values of Rice factor K and means µP1 and µP2. After that analytical expression (22) for pIR
out is validated in
Fig. 5. Monte-Carlo simulation results matching closely with
Position (x coordinate) of relay (m)
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 O u ta ge p ro b ab il it y 10−4 10−3 10−2 10−1 100 N s= 10, dAS= 1 m Ns= 20, dAS= 1 m Ns= 10, dAS= 2 m Ns= 20, dAS= 2 m Simulation No relaying Ns= 10, dAS= 2 m Ns= 10, dAS= 1 m Ns= 20, dAS= 1 m Ns= 20, dAS= 2 m
Fig. 5: Variation of pout in IR and NR with xR, dAS, Ns. R0 is
